With leaders from the 19 single currency nations gathering in Brussels, financial officials gave a tentative endorsement to new Greek proposals for spending cuts and reforms they would make in exchange for billions of euros in fresh loans.
"It's an opportunity to get that deal this week," said Jeroen Dijsselbloem, the Dutchman who chaired an emergency meeting of eurozone finance ministers ahead of the EU summit.
He said technical experts who are evaluating the Greek plan believe it "is broad and comprehensive but they really need to look at the specifics to see whether it adds up in fiscal terms."
The news gave a boost to stock markets. Athens shares, which had lost much of its strong gains on the news that there has been no definitive deal at a meeting on Monday, rebounded on the positive comments. It closed 9 per cent higher. The Stoxx 50 index of top European shares was somewhat less volatile and closed 4.1 per cent higher.
The need for a deal could not be more pressing. Greece must pay 1.6 billion euros ($1.8 billion) to the International Monetary Fund in just over a week, on June 30. Further payments are due in July and August, and the leftwing government in Athens does not have the money to pay them.
Negotiations on the reforms that Greece must undertake to secure its financial future have dragged on since February.
A debt default by Greece could destabilize its banks — Greeks are already withdrawing increasingly large amounts of money — and could in a worst case scenario cause the country to have to leave the euro.
That would be hugely painful for Greeks but experts are more divided about its effects on Europe and the world economy. Several European countries have said publicly they are getting prepared for the possibility.
Despite the urgency of the situation, German Chancellor Angela Merkel and other European officials warned against focusing too much on Monday's summit.
"There are still a lot of days left to come to a decision," Merkel said before heading to Brussels.
European Commission President Jean-Claude Juncker, who has been a key player in the efforts to broker a deal, said that "we are on the right path, but we still have some way to go."
Leaders of the 28-nation European Union have a two-day summit starting Thursday and it is hoped that a full agreement can be reached then.
Greece's economy minister, Giorgios Stathakis, was upbeat, saying that Athens' latest offer effectively breaks the deadlock with international lenders.
"They have accepted that the new proposal of the Greek government is a proper framework on which to work," he told Britain's BBC.
Stathakis said the proposals included new taxes on business and the wealthy but no further cuts in pensions or public sector salaries, which was still a "red line" for Greece.
However, Athens will make tougher rules on early retirement and shift some categories of goods to a higher sales tax bracket, including hotels and certain foods. Emergency bailout taxes that had been imposed will remain, even though Tsipras had pledged to phase them out.
Despite the more upbeat mood in markets, tension was palpable in Greece, where people have flocked to cash machines to withdraw money. The concern is that a debt default by Greece could destabilize the country enough that it might eventually have to leave the euro.
"Everyone's going (to the banks) to take money," said Yannis Nikolopoulos in Athens. "If the banks shut it'll be a problem to go shopping and that sort of thing." Without a deal, he said, "we're doomed."
To support Greek banks in the face of growing money withdrawals, the European Central Bank increased the amount of emergency credit it allows the banks to draw on, officials said.
Reports indicate Greeks withdrew about 4 billion euros last week.
The current talks centre on releasing the last 7.2 billion euros in the country's bailout program, which expires at the end of the month.
Since coming to power in January, the new government has refused to make more budget austerity measures, which it blames for devastating the economy. It has since softened its approach, but it remains reluctant to take all steps creditors demand.
Derek Gatopoulos in Brussels and Nathalie Savaricas in Athens contributed to this report.